On the Spread of Random Interleaver

TL;DR

This paper analyzes the probability distribution of interleavers with a specific spread in coding theory, showing that as blocklength increases, the probability of having spread two approaches approximately 86.47%, and provides bounds for higher spreads.

Contribution

It derives exact counts and probabilities for interleavers with spread two and establishes asymptotic bounds for higher spreads as blocklength grows.

Findings

Probability of spread two approaches 0.8647 as blocklength increases

Derived lower bounds for probability of spread at least s

Asymptotic convergence of bounds to e^{-2(s-2)^2}

Abstract

For a given blocklength we determine the number of interleavers which have spread equal to two. Using this, we find out the probability that a randomly chosen interleaver has spread two. We show that as blocklength increases, this probability increases but very quickly converges to the value $1-e^{-2} \approx 0.8647$. Subsequently, we determine a lower bound on the probability of an interleaver having spread at least $s$. We show that this lower bound converges to the value $e^{-2(s-2)^{2}}$, as the blocklength increases.